Geometric series is very important in analysis. It denition is
\( \sum _{n=0}^{\infty}(z)^n \)
Calculate this infinite sum is easier as seems if we note following
\( S_{n} = \sum _{n=0}^{\infty}(z)^n = 1 + z + z^2 + ... + z^n \Rightarrow zS_{n}= z + z^2 + ... + z^{n+1} \)
this implies
\( S_{n}(1-z) = 1 + z + z^2 + ... + z^n - (z + z^2 + ... + z^{n+1}) = 1 + z^{n+1}\)
thus
\(S_{n} = \frac{1 + z^{n+1}}{1-z}\)
Now is easy taking limit when \( n\mapsto \infty \)
\( \sum _{n=0}^{\infty}(z)^n = \frac{1}{1-z}\)
Geometric Series
Sum of geometric series
Convergence radius of geometric series
We apply the ratio test
\( \lim_{n\to \infty} |\frac{S_{n+1}}{Z_{n}}| = \lim_{n\to \infty} |\frac{x^{n+1}}{z{n}}| = |z| \)
Thus, geometric series converges if |z| < 1 and diverges if |z| > 1